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\title{Enumerating possible combinations of subgraphs with a connected union}
\author{}

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\maketitle

\section{Problem statement}
We are considering the following problem. Given a graph $G$ with unique node labelings, and the set of all it's subgraphs $G'$ (both connected), 
we want to find the set $S \subset P(G')$ such that for a given combination of subgraphs $x=\{G_1, ..., G_k\} \in S$, we have that:
\begin{equation}
  \forall G_i \in x, \exists G_j \in x, i \neq j ,  \quad G_i \cap G_j \neq \emptyset 
\end{equation}

This may also be reformulated as:
\begin{equation}
  (1) \iff \bigcup_{i=1}^{k}G_{i} \text{ is connected}
\end{equation}

That is, $S$ is the set of all those combinations of subgraphs such that for any subgraph in a combination, there is
atleast some other subgraph in the same combination which is non-disjoint (when regarding the vertex set).
This will ensure that the union of those subgraphs is connected.

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